Integrand size = 16, antiderivative size = 32 \[ \int \csc (c-b x) \sin ^2(a+b x) \, dx=\frac {\cos (2 a+c+b x)}{b}+\frac {\text {arctanh}(\cos (c-b x)) \sin ^2(a+c)}{b} \] Output:
cos(b*x+2*a+c)/b+arctanh(cos(b*x-c))*sin(a+c)^2/b
Time = 0.10 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \csc (c-b x) \sin ^2(a+b x) \, dx=\frac {\cos (2 a+c+b x)+\left (\log \left (\cos \left (\frac {1}{2} (c-b x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c-b x)\right )\right )\right ) \sin ^2(a+c)}{b} \] Input:
Integrate[Csc[c - b*x]*Sin[a + b*x]^2,x]
Output:
(Cos[2*a + c + b*x] + (Log[Cos[(c - b*x)/2]] - Log[Sin[(c - b*x)/2]])*Sin[ a + c]^2)/b
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sin ^2(a+b x) \csc (c-b x) \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \sin ^2(a+b x) \csc (c-b x)dx\) |
Input:
Int[Csc[c - b*x]*Sin[a + b*x]^2,x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 127, normalized size of antiderivative = 3.97
method | result | size |
risch | \(\frac {\ln \left ({\mathrm e}^{i \left (a +c \right )}+{\mathrm e}^{i \left (b x +a \right )}\right )}{2 b}-\frac {\ln \left ({\mathrm e}^{i \left (a +c \right )}+{\mathrm e}^{i \left (b x +a \right )}\right ) \cos \left (2 a +2 c \right )}{2 b}-\frac {\ln \left (-{\mathrm e}^{i \left (a +c \right )}+{\mathrm e}^{i \left (b x +a \right )}\right )}{2 b}+\frac {\ln \left (-{\mathrm e}^{i \left (a +c \right )}+{\mathrm e}^{i \left (b x +a \right )}\right ) \cos \left (2 a +2 c \right )}{2 b}+\frac {\cos \left (b x +2 a +c \right )}{b}\) | \(127\) |
default | \(-\frac {\frac {2 \left (\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )+2 \sin \left (a \right ) \sin \left (c \right )-2 \cos \left (a \right ) \cos \left (c \right )}{\left (\cos \left (a \right )^{2} \cos \left (c \right )^{2}+\sin \left (c \right )^{2} \cos \left (a \right )^{2}+\cos \left (c \right )^{2} \sin \left (a \right )^{2}+\sin \left (a \right )^{2} \sin \left (c \right )^{2}\right ) \left (1+\tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}\right )}+\frac {8 \left (\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right )^{2} \arctan \left (\frac {2 \left (\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )-2 \sin \left (a \right ) \sin \left (c \right )+2 \cos \left (a \right ) \cos \left (c \right )}{2 \sqrt {-\cos \left (c \right )^{2} \sin \left (a \right )^{2}-\cos \left (a \right )^{2} \cos \left (c \right )^{2}-\sin \left (a \right )^{2} \sin \left (c \right )^{2}-\sin \left (c \right )^{2} \cos \left (a \right )^{2}}}\right )}{\left (4 \cos \left (c \right )^{2} \sin \left (a \right )^{2}+4 \cos \left (a \right )^{2} \cos \left (c \right )^{2}+4 \sin \left (a \right )^{2} \sin \left (c \right )^{2}+4 \sin \left (c \right )^{2} \cos \left (a \right )^{2}\right ) \sqrt {-\cos \left (c \right )^{2} \sin \left (a \right )^{2}-\cos \left (a \right )^{2} \cos \left (c \right )^{2}-\sin \left (a \right )^{2} \sin \left (c \right )^{2}-\sin \left (c \right )^{2} \cos \left (a \right )^{2}}}}{b}\) | \(278\) |
Input:
int(-csc(b*x-c)*sin(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/2/b*ln(exp(I*(a+c))+exp(I*(b*x+a)))-1/2/b*ln(exp(I*(a+c))+exp(I*(b*x+a)) )*cos(2*a+2*c)-1/2/b*ln(-exp(I*(a+c))+exp(I*(b*x+a)))+1/2/b*ln(-exp(I*(a+c ))+exp(I*(b*x+a)))*cos(2*a+2*c)+cos(b*x+2*a+c)/b
Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (33) = 66\).
Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.69 \[ \int \csc (c-b x) \sin ^2(a+b x) \, dx=\frac {2 \, \cos \left (b x + a\right ) \cos \left (a + c\right ) - {\left (\cos \left (a + c\right )^{2} - 1\right )} \log \left (\frac {\cos \left (b x + a\right ) \cos \left (a + c\right ) + \sin \left (b x + a\right ) \sin \left (a + c\right ) + 1}{\cos \left (a + c\right ) + 1}\right ) + {\left (\cos \left (a + c\right )^{2} - 1\right )} \log \left (-\frac {\cos \left (b x + a\right ) \cos \left (a + c\right ) + \sin \left (b x + a\right ) \sin \left (a + c\right ) - 1}{\cos \left (a + c\right ) + 1}\right ) - 2 \, \sin \left (b x + a\right ) \sin \left (a + c\right )}{2 \, b} \] Input:
integrate(-csc(b*x-c)*sin(b*x+a)^2,x, algorithm="fricas")
Output:
1/2*(2*cos(b*x + a)*cos(a + c) - (cos(a + c)^2 - 1)*log((cos(b*x + a)*cos( a + c) + sin(b*x + a)*sin(a + c) + 1)/(cos(a + c) + 1)) + (cos(a + c)^2 - 1)*log(-(cos(b*x + a)*cos(a + c) + sin(b*x + a)*sin(a + c) - 1)/(cos(a + c ) + 1)) - 2*sin(b*x + a)*sin(a + c))/b
Leaf count of result is larger than twice the leaf count of optimal. 690 vs. \(2 (27) = 54\).
Time = 10.61 (sec) , antiderivative size = 3216, normalized size of antiderivative = 100.50 \[ \int \csc (c-b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(-csc(b*x-c)*sin(b*x+a)**2,x)
Output:
2*Piecewise((0, Eq(b, 0) & Eq(c, 0)), (-sin(b*x)/b, Eq(c, 0)), (0, Eq(b, 0 )), (2*log(-tan(c/2) + tan(b*x/2))*tan(c/2)**3*tan(b*x/2)**2/(b*tan(c/2)** 4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan( c/2)**2 + b*tan(b*x/2)**2 + b) + 2*log(-tan(c/2) + tan(b*x/2))*tan(c/2)**3 /(b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2) **2 + 2*b*tan(c/2)**2 + b*tan(b*x/2)**2 + b) - 2*log(-tan(c/2) + tan(b*x/2 ))*tan(c/2)*tan(b*x/2)**2/(b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2 *b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan(c/2)**2 + b*tan(b*x/2)**2 + b) - 2* log(-tan(c/2) + tan(b*x/2))*tan(c/2)/(b*tan(c/2)**4*tan(b*x/2)**2 + b*tan( c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan(c/2)**2 + b*tan(b*x/2)** 2 + b) - 2*log(tan(b*x/2) + 1/tan(c/2))*tan(c/2)**3*tan(b*x/2)**2/(b*tan(c /2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b *tan(c/2)**2 + b*tan(b*x/2)**2 + b) - 2*log(tan(b*x/2) + 1/tan(c/2))*tan(c /2)**3/(b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan( b*x/2)**2 + 2*b*tan(c/2)**2 + b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2) + 1/ tan(c/2))*tan(c/2)*tan(b*x/2)**2/(b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2) **4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan(c/2)**2 + b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2) + 1/tan(c/2))*tan(c/2)/(b*tan(c/2)**4*tan(b*x/2)**2 + b*tan(c/2)**4 + 2*b*tan(c/2)**2*tan(b*x/2)**2 + 2*b*tan(c/2)**2 + b*tan( b*x/2)**2 + b) + 2*tan(c/2)**4*tan(b*x/2)/(b*tan(c/2)**4*tan(b*x/2)**2 ...
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (33) = 66\).
Time = 0.05 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.62 \[ \int \csc (c-b x) \sin ^2(a+b x) \, dx=-\frac {{\left (\cos \left (2 \, a + 2 \, c\right ) - 1\right )} \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) - {\left (\cos \left (2 \, a + 2 \, c\right ) - 1\right )} \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) - 4 \, \cos \left (b x + 2 \, a + c\right )}{4 \, b} \] Input:
integrate(-csc(b*x-c)*sin(b*x+a)^2,x, algorithm="maxima")
Output:
-1/4*((cos(2*a + 2*c) - 1)*log(cos(b*x)^2 + 2*cos(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(c) + sin(c)^2) - (cos(2*a + 2*c) - 1)*log(cos (b*x)^2 - 2*cos(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(c) + sin(c)^2) - 4*cos(b*x + 2*a + c))/b
Leaf count of result is larger than twice the leaf count of optimal. 684 vs. \(2 (33) = 66\).
Time = 0.13 (sec) , antiderivative size = 684, normalized size of antiderivative = 21.38 \[ \int \csc (c-b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(-csc(b*x-c)*sin(b*x+a)^2,x, algorithm="giac")
Output:
-2*(2*(tan(1/2*a)^4*tan(1/2*c)^2 + 2*tan(1/2*a)^3*tan(1/2*c)^3 + tan(1/2*a )^2*tan(1/2*c)^4 - 2*tan(1/2*a)^3*tan(1/2*c) - 4*tan(1/2*a)^2*tan(1/2*c)^2 - 2*tan(1/2*a)*tan(1/2*c)^3 + tan(1/2*a)^2 + 2*tan(1/2*a)*tan(1/2*c) + ta n(1/2*c)^2)*log(abs(tan(1/2*b*x - 1/2*c)))/(tan(1/2*a)^4*tan(1/2*c)^4 + 2* tan(1/2*a)^4*tan(1/2*c)^2 + 2*tan(1/2*a)^2*tan(1/2*c)^4 + tan(1/2*a)^4 + 4 *tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*c)^4 + 2*tan(1/2*a)^2 + 2*tan(1/2*c)^ 2 + 1) - (4*tan(1/2*b*x - 1/2*c)*tan(1/2*a)^4*tan(1/2*c)^3 + 4*tan(1/2*b*x - 1/2*c)*tan(1/2*a)^3*tan(1/2*c)^4 + tan(1/2*a)^4*tan(1/2*c)^4 - 4*tan(1/ 2*b*x - 1/2*c)*tan(1/2*a)^4*tan(1/2*c) - 24*tan(1/2*b*x - 1/2*c)*tan(1/2*a )^3*tan(1/2*c)^2 - 6*tan(1/2*a)^4*tan(1/2*c)^2 - 24*tan(1/2*b*x - 1/2*c)*t an(1/2*a)^2*tan(1/2*c)^3 - 16*tan(1/2*a)^3*tan(1/2*c)^3 - 4*tan(1/2*b*x - 1/2*c)*tan(1/2*a)*tan(1/2*c)^4 - 6*tan(1/2*a)^2*tan(1/2*c)^4 + 4*tan(1/2*b *x - 1/2*c)*tan(1/2*a)^3 + tan(1/2*a)^4 + 24*tan(1/2*b*x - 1/2*c)*tan(1/2* a)^2*tan(1/2*c) + 16*tan(1/2*a)^3*tan(1/2*c) + 24*tan(1/2*b*x - 1/2*c)*tan (1/2*a)*tan(1/2*c)^2 + 36*tan(1/2*a)^2*tan(1/2*c)^2 + 4*tan(1/2*b*x - 1/2* c)*tan(1/2*c)^3 + 16*tan(1/2*a)*tan(1/2*c)^3 + tan(1/2*c)^4 - 4*tan(1/2*b* x - 1/2*c)*tan(1/2*a) - 6*tan(1/2*a)^2 - 4*tan(1/2*b*x - 1/2*c)*tan(1/2*c) - 16*tan(1/2*a)*tan(1/2*c) - 6*tan(1/2*c)^2 + 1)/((tan(1/2*a)^4*tan(1/2*c )^4 + 2*tan(1/2*a)^4*tan(1/2*c)^2 + 2*tan(1/2*a)^2*tan(1/2*c)^4 + tan(1/2* a)^4 + 4*tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*c)^4 + 2*tan(1/2*a)^2 + 2*...
Time = 1.68 (sec) , antiderivative size = 223, normalized size of antiderivative = 6.97 \[ \int \csc (c-b x) \sin ^2(a+b x) \, dx=\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-c\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}}{2\,b}+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+c\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}}{2\,b}-\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}\,\ln \left (-\frac {{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}-1\right )}^2\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left (-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}\,2{}\mathrm {i}+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\mathrm {e}}^{c\,4{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}{2}\right )\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}-1\right )}^2}{4\,b}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}\,\ln \left (\frac {{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}-1\right )}^2\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left (-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}\,2{}\mathrm {i}+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\mathrm {e}}^{c\,4{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}{2}\right )\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}-1\right )}^2}{4\,b} \] Input:
int(sin(a + b*x)^2/sin(c - b*x),x)
Output:
exp(- a*2i - c*1i - b*x*1i)/(2*b) + exp(a*2i + c*1i + b*x*1i)/(2*b) - (exp (- a*2i - c*2i)*log(- ((exp(a*2i)*exp(c*2i) - 1)^2*1i)/2 - (exp(-c*1i)*exp (b*x*1i)*(exp(a*4i)*exp(c*4i)*1i - exp(a*2i)*exp(c*2i)*2i + 1i))/2)*(exp(a *2i + c*2i) - 1)^2)/(4*b) + (exp(- a*2i - c*2i)*log(((exp(a*2i)*exp(c*2i) - 1)^2*1i)/2 - (exp(-c*1i)*exp(b*x*1i)*(exp(a*4i)*exp(c*4i)*1i - exp(a*2i) *exp(c*2i)*2i + 1i))/2)*(exp(a*2i + c*2i) - 1)^2)/(4*b)
\[ \int \csc (c-b x) \sin ^2(a+b x) \, dx=-\left (\int \csc \left (b x -c \right ) \sin \left (b x +a \right )^{2}d x \right ) \] Input:
int(-csc(b*x-c)*sin(b*x+a)^2,x)
Output:
- int(csc(b*x - c)*sin(a + b*x)**2,x)